Image setting in UV Project Modifier
So what actually the UV Project modifier does is only manipulates UV coordinates. Without an UV map it won't work. The Override option will work only in Multitexture, it can't override material textures, only textures displayed on mesh faces.

All the stuff about view-port visibility and what renders is nor related to this modifier but rather to how blender works.


Multi-texture Blender Internal:

In view-port Blender is able to preview textures without any materials applied - just the ones in the UV/Image editor. In render it uses materials. That's why view-port and render differs. The UV mapping is altered with the UV Project modifier - it is altered for both render and view-port.
Cycles and GLSL Blender Internal:

In view-port Blender previews material textures. If a texture is not in a material, you can't see it on the model.

For Blender Internal materials there is UV Project option to ensure correct UV interpolation for UV Project modifier. For both GLSL and Cycles the material mapping should be UV:






If you setup a material for Multitexture the same way like for GLSL it will work in render.



Everything is correct in your .blend, it's just that you expected the UV Project modifier to override material textures, which is not the case, and you need to use the "AB" texture instead of "B" in your shaders.